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Abstract

Let $M$ be an $N$-function satisfying the $\Delta_2$-condition, and let $\omega, \varphi$ be two other functions, with $\omega\ge 0$. We study Hardy-type inequalities $$ \int_{{\mathbb R}_+} M(\omega (x)|u(x)|) \exp (-\varphi (x))\,dx \le C\int_{{\mathbb R}_+} M(|u'(x)|) \exp (-\varphi (x))\,dx, $$ where $u$ belongs to some set ${\cal R }$ of locally absolutely continuous functions containing $C_0^\infty ({\mathbb R}_+)$. We give sufficient conditions on the triple $(\omega,\varphi,M)$ for such inequalities to be valid for all $u$ from a given set ${\cal R}$. The set ${\cal R}$ may be smaller than the set of Hardy transforms. Bounds for constants are also given, yielding classical Hardy inequalities with best constants.